Editing Euler equations (fluid dynamics) (section)

==Incompressible Euler equations with constant and uniform density==
In convective form (i.e., the form with the [[convective derivative|convective operator]] made explicit in the [[Cauchy momentum equation|momentum equation]]), the incompressible Euler equations in case of density constant in time and uniform in space are:{{sfn|Hunter|2006|p=}}

{{Equation box 1
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|title='''Incompressible Euler equations with constant and uniform density'''<br/>(''convective or Lagrangian form'')
|equation=<math>\begin{align} 
  {D\mathbf{u} \over Dt} &= -\nabla w + \mathbf{g} \\
  \nabla\cdot \mathbf{u} &= 0
\end{align}</math>
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where:
*<math>\mathbf u</math> is the [[flow velocity]] [[Vector (geometric)|vector]], with components in an ''N''-dimensional space <math>u_1, u_2, \dots, u_N</math>,
*<math>\frac{D\boldsymbol\Phi}{Dt} = \frac{\partial\boldsymbol\Phi}{\partial t} + \mathbf v \cdot \nabla \boldsymbol\Phi</math>, for a generic function (or field) <math>\boldsymbol\Phi</math> denotes its [[material derivative]] in time with respect to the advective field <math>\mathbf v</math> and
*<math>\nabla w</math> is the [[gradient]] of the specific (with the sense of ''per unit mass'') [[thermodynamic work]], the internal [[Linear differential equation|source term]], and
*<math>\nabla \cdot \mathbf u</math> is the flow velocity [[divergence]].
* <math>\mathbf{g}</math> represents [[body force|body acceleration]]s (per unit mass) acting on the continuum, for example [[gravity]], [[inertial acceleration]]s, [[electric field]] acceleration, and so on.

The first equation is the [[Cauchy momentum equation|Euler momentum equation]] with uniform density (for this equation it could also not be constant in time). By expanding the [[material derivative]], the equations become:
<math display="block">\begin{align} 
  {\partial\mathbf{u} \over \partial t} + (\mathbf{u} \cdot \nabla) \mathbf{u} &= -\nabla w + \mathbf{g},\\
                                                     \nabla \cdot \mathbf{u} &= 0.
\end{align}</math>

In fact for a flow with uniform density <math>\rho_0</math> the following identity holds:
<math display="block">\nabla w \equiv \nabla \left(\frac p {\rho_0} \right) = \frac 1 {\rho_0} \nabla p,</math>
where <math>p</math> is the mechanic [[pressure]]. The second equation is the [[incompressible flow|incompressible constraint]], stating the flow velocity is a [[solenoidal field]] (the order of the equations is not causal, but underlines the fact that the incompressible constraint is not a degenerate form of the [[continuity equation]], but rather of the energy equation, as it will become clear in the following). Notably, the [[continuity equation]] would be required also in this incompressible case as an additional third equation in case of density varying in time ''or'' varying in space. For example, with density nonuniform in space but constant in time, the continuity equation to be added to the above set would correspond to:
<math display="block">\frac{\partial \rho}{\partial t} = 0.</math>

So the case of constant ''and'' uniform density is the only one not requiring the continuity equation as additional equation regardless of the presence or absence of the incompressible constraint. In fact, the case of incompressible Euler equations with constant and uniform density discussed here is a [[toy model]] featuring only two simplified equations, so it is ideal for didactical purposes even if with limited physical relevance.

The equations above thus represent respectively [[conservation of mass]] (1 scalar equation) and [[conservation of momentum|momentum]] (1 vector equation containing <math>N</math> scalar components, where <math>N</math> is the physical dimension of the space of interest). Flow velocity and pressure are the so-called ''physical variables''.{{sfn|Toro|1999|p= 24}}

In a coordinate system given by <math>\left(x_1, \dots, x_N\right)</math> the velocity and external force vectors <math>\mathbf u</math> and <math>\mathbf g</math> have components <math>(u_1,\dots, u_N)</math> and <math>\left(g_1, \dots, g_N\right)</math>, respectively. Then the equations may be expressed in subscript notation as:

<math display="block">\begin{align} 
  {\partial u_i \over \partial t} + \sum_{j=1}^N {\partial \left(u_i u_j + w\delta_{ij}\right) \over \partial x_j} &= g_i,\\
  \sum_{i=1}^N {\partial u_i \over \partial x_i} &= 0.
\end{align}</math>

where the <math>i</math> and <math>j</math> subscripts label the ''N''-dimensional space components, and <math>\delta_{ij}</math> is the [[Kroenecker delta]]. The use of [[Einstein notation]] (where the sum is implied by repeated indices instead of [[Summation#Capital-sigma notation|sigma notation]]) is also frequent.

===Properties===
Although Euler first presented these equations in 1755, many fundamental questions or concepts  about them remain unanswered.

In three space dimensions, in certain simplified scenarios, the Euler equations produce singularities.<ref>{{Cite journal |last=Elgindi |first=Tarek M. |date=2021-11-01 |title=Finite-time singularity formation for $C^{1,\alpha}$ solutions to the incompressible Euler equations on $\mathbb{R}^3$ |url=https://projecteuclid.org/journals/annals-of-mathematics/volume-194/issue-3/Finite-time-singularity-formation-for-C1alpha-solutions-to-the-incompressible/10.4007/annals.2021.194.3.2.full |journal=Annals of Mathematics |volume=194 | arxiv = 1904.04795 |issue=3 |doi=10.4007/annals.2021.194.3.2 |issn=0003-486X}}</ref>

Smooth solutions of the free (in the sense of without source term: g=0) equations satisfy the conservation of specific kinetic energy:
<math display="block">{\partial \over\partial t} \left(\frac{1}{2} u^2 \right) + \nabla \cdot \left(u^2 \mathbf u + w \mathbf u\right) = 0.</math>

In the one-dimensional case without the source term (both pressure gradient and external force), the momentum equation becomes the inviscid [[Burgers' equation]]:
<math display="block">{\partial u \over\partial t}+ u {\partial u \over\partial x} = 0.</math>

This model equation gives many insights into Euler equations.

===Nondimensionalisation===
{{See also|Cauchy momentum equation#Nondimensionalisation}}
{{Unreferenced section|date=April 2021}}
In order to make the equations dimensionless, a characteristic length <math>r_0</math>, and a characteristic velocity <math>u_0</math>, need to be defined. These should be chosen such that the dimensionless variables are all of order one. The following dimensionless variables are thus obtained:
<math display="block">\begin{align}
       u^* & \equiv \frac{u}{u_0}, &
       r^* & \equiv \frac{r}{r_0}, \\[5pt]
       t^* & \equiv \frac{u_0}{r_0} t, &
       p^* & \equiv \frac{w}{u_0^2}, \\[5pt]
  \nabla^* & \equiv r_0 \nabla.
\end{align}</math>
and of the field [[unit vector]]:
<math display="block">\hat{\mathbf g}\equiv \frac {\mathbf g} g.</math>

Substitution of these inversed relations in Euler equations, defining the [[Froude number]], yields (omitting the * at apix):
{{Equation box 1
|indent=:
|title='''Incompressible Euler equations with constant and uniform density'''<br/>(''nondimensional form'')
|equation=<math>\begin{align} 
   {D\mathbf{u} \over Dt} &= -\nabla w + \frac{1}{\mathrm{Fr}} \hat{\mathbf{g}}\\
  \nabla \cdot \mathbf{u} &= 0
\end{align}</math>
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Euler equations in the Froude limit (no external field) are named free equations and are conservative. The limit of high Froude numbers (low external field) is thus notable and can be studied with [[perturbation theory]].

===Conservation form===
{{See also|Conservation equation|}}
The conservation form emphasizes the mathematical properties of Euler equations, and especially the contracted form is often the most convenient one for [[computational fluid dynamics]] simulations. Computationally, there are some advantages in using the conserved variables. This gives rise to a large class of numerical methods
called conservative methods.{{sfn|Toro|1999|p= 24}}

The '''free Euler equations are conservative''', in the sense they are equivalent to a conservation equation:
<math display="block">
\frac{\partial \mathbf y}{\partial t}+ \nabla \cdot \mathbf F ={\mathbf 0},
</math>
or simply in Einstein notation:
<math display="block">
\frac{\partial y_j}{\partial t}+
\frac{\partial f_{ij}}{\partial r_i}= 0_i,
</math>
where the conservation quantity <math>\mathbf y</math> in this case is a vector, and <math>\mathbf F</math> is a [[flux]] matrix. This can be simply proved.

{{hidden
|Demonstration of the conservation form
|First, the following identities hold:
<math display="block">\nabla \cdot (w \mathbf I) = \mathbf I \cdot \nabla w + w \nabla \cdot \mathbf I = \nabla w </math>
<math display="block">\mathbf u \cdot \nabla \cdot \mathbf u = \nabla \cdot (\mathbf u \otimes \mathbf u)</math>
where <math>\otimes</math> denotes the [[outer product]]. The same identities expressed in [[Einstein notation]] are:
<math display="block">\partial_i\left(w \delta_{ij}\right) = \delta_{ij} \partial_i w + w \partial_i \delta_{ij} = \delta_{ij} \partial_i w =  \partial_j w</math>
<math display="block">u_j \partial_i u_i = \partial_i \left(u_i u_j\right)</math>
where {{mvar|I}} is the [[identity matrix]] with dimension {{mvar|N}} and {{mvar|δ<sub>ij</sub>}} its general element, the Kroenecker delta.

Thanks to these vector identities, the incompressible Euler equations with constant and uniform density and without external field can be put in the so-called ''conservation'' (or Eulerian) differential form, with vector notation:
<math display="block">\left\{\begin{align} 
  {\partial\mathbf{u} \over \partial t} + \nabla \cdot \left(\mathbf{u} \otimes \mathbf{u} + w\mathbf{I}\right) &= \mathbf{0} \\
  {\partial 0 \over \partial t} + \nabla \cdot \mathbf{u} &= 0,
\end{align}\right.</math>
or with Einstein notation:
<math display="block">\left\{\begin{align} 
  \partial_t u_j + \partial_i \left(u_i u_j + w \delta_{ij}\right) &= 0 \\
                                     \partial_t 0 + \partial_j u_j &= 0,
\end{align}\right.</math>

Then '''incompressible''' Euler equations with uniform density have conservation variables:
<math display="block">
\mathbf y = \begin{pmatrix}\mathbf u \\ 0 \end{pmatrix}; \qquad
\mathbf F = \begin{pmatrix}\mathbf u \otimes \mathbf u + w \mathbf I \\ \mathbf u \end{pmatrix}.
</math>

Note that in the second component u is by itself a vector, with length N, so y has length N+1 and F has size N(N+1). In 3D for example y has length 4, I has size 3×3 and F has size 4×3, so the explicit forms are:
<math display="block">
{\mathbf y}=\begin{pmatrix}  u_1 \\ u_2  \\ u_3 \\0 \end{pmatrix}; \quad
{\mathbf F}=\begin{pmatrix}
u_1^2 + w &  u_1u_2 &  u_1u_3
\\  u_2 u_1 & u_2^2 + w &  u_2 u_3
\\ u_3 u_1 &  u_3 u_2 &  u_3^2 + w 
\\ u_1 & u_2 & u_3 \end{pmatrix}.
</math>

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At last Euler equations can be recast into the particular equation:

{{Equation box 1
|indent=:
|title='''Incompressible Euler equation(s) with constant and uniform density'''<br/>(''conservation or Eulerian form'')
|equation=<math>
\frac {\partial}{\partial t}\begin{pmatrix} \mathbf u \\ 0 \end{pmatrix} + \nabla \cdot \begin{pmatrix}\mathbf u \otimes \mathbf u + w \mathbf I \\ \mathbf u \end{pmatrix} = \begin{pmatrix}\mathbf g \\ 0\end{pmatrix}
</math>
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===Spatial dimensions===
For certain problems, especially when used to analyze compressible flow in a duct or in case the flow is cylindrically or spherically symmetric, the one-dimensional Euler equations are a useful first approximation. Generally, the Euler equations are solved by [[Riemann]]'s [[method of characteristics]]. This involves finding curves in plane of independent variables (i.e., <math>x</math> and <math>t</math>) along which [[partial differential equation]]s (PDEs) degenerate into [[ordinary differential equation]]s (ODEs). [[Numerical analysis|Numerical solutions]] of the Euler equations rely heavily on the method of characteristics.